Quantum-Inspired Evolutionary Algorithm for Continuous Space Optimization Based on Multiple Chains Encoding Method of Quantum Bits

Pseudocode 3

Begin

𝑡=0

(1) Initialize the colony 𝑄(𝑡)=(𝑝_𝑖1,𝑝_𝑖2,𝑝_𝑖3,𝑝_𝑖4).

(2) Construct 𝑋(𝑡)=(𝑋^𝑡_𝑖1,𝑋^𝑡_𝑖2,𝑋^𝑡_𝑖3,𝑋^𝑡_𝑖4) by solution space transformation.

(3) Obtain the optimum solution of the current 𝐵𝑒𝑠𝑡𝑋 among 𝑋(𝑡) and the current  optimum chromosome 𝐵𝑒𝑠𝑡𝐶 among 𝑄(𝑡)

  by evaluating 𝑋(𝑡).

(4) Store 𝐵𝑒𝑠𝑡𝑋 into 𝐺𝑋 and 𝐵𝑒𝑠𝑡𝐶 into 𝐺𝐶.

  While (not termination-condition)

   Begin

   𝑡=𝑡+1

(5) Calculate 𝑄(𝑡) by updating and mutating the states of 𝑄(𝑡−1).

   Determine 𝑋(𝑡)=(𝑋^𝑡_𝑖1,𝑋^𝑡_𝑖2,𝑋^𝑡_𝑖3,𝑋^𝑡_𝑖4) by solution space transformation.

(6) Obtain the optimum solution of the current 𝐵𝑋 among 𝑋(𝑡) and the current optimum chromosome 𝐵𝐶 among 𝑄(𝑡)

    by evaluating 𝑋(𝑡).

(7) If 𝑓𝑖𝑡(𝐵𝑋)<𝑓𝑖𝑡(𝐺𝑋)

   𝐵𝑒𝑠𝑡𝑋=𝐺𝑋; 𝐵𝑒𝑠𝑡𝐶=𝐺𝐶

   Otherwise

   𝐺𝑋=𝐵𝑒𝑠𝑡𝑋; 𝐺𝐶=𝐵𝑒𝑠𝑡𝐶

   End

  End

End

(8) The procedure of FCQIEA can be summarized as follows:

   Step  1. Initialize the population. Let the current generation 𝑡=0; generate an initial

population 𝑄(𝑡)=𝑞^𝑡₁,𝑞^𝑡₂,…,𝑞^𝑡_𝑚, which has 𝑚 individual qubits. Set the magnitude of the rotational angle Δ𝛽=𝛽₀,

Δ𝜑=𝜑₀, and |Δ𝜃|=𝜃₀, respectively. Set 𝑝_𝑚 as the mutation and 𝑀𝑎𝑥_𝑔𝑒𝑛 as the maximum generation.

   Step  2. Transform the solution space. Four approximate solutions in each chromosome are transformed from the unit space

𝐼^𝑛=[−1,1]^𝑛 to the solution space Ω of the continuous optimization problem (1); thus, the set of approximate solution

𝑋(𝑡) can be obtained.

   Step  3. Compute the fitness. By computing the fitness of 4𝑚 approximate solutions, obtain the best solution 𝐵𝑒𝑠𝑡𝑋 in the

current solution and the best chromosome 𝐵𝑒𝑠𝑡𝐶 in the current chromosome. Store 𝐵𝑒𝑠𝑡𝑋 as the global optimum solution 𝐺𝑋

and store 𝐵𝑒𝑠𝑡𝐶 as the global optimum chromosome 𝐺𝐶.

   Step  4. Set 𝑡=𝑡+1. Update and mutate 𝑄(𝑡−1). Calculate the new population 𝑄(𝑡).

   Step  5. Transform the solution space again and obtain a set of approximate solution 𝑋(𝑡).

   Step  6. By computing the fitness of 𝑋(𝑡), determine the current optimum solution 𝐵𝑒𝑠𝑡𝑋 and the current optimum.

chromosome 𝐵𝑒𝑠𝑡𝐶 If 𝑓𝑖𝑡(𝐵𝑒𝑠𝑡𝑋)<𝑓𝑖𝑡(𝐺𝑋), then update the current optimum solution 𝐵𝑒𝑠𝑡𝑋=𝐺𝑋; at the same time,

update the current optimum chromosome 𝐵𝑒𝑠𝑡𝐶=𝐺𝐶 to avoid population degradation. Otherwise, let 𝐺𝑋=𝐵𝑒𝑠𝑡𝑋

and 𝐺𝐶=𝐵𝑒𝑠𝑡𝐶 so that the algorithm approaches the global optimum solution.

   Step  7. If the algorithm does not converge and if 𝑡<𝑀𝑎𝑥_𝑔𝑒𝑛, then go back to Step 4 until the algorithm

becomes convergent or until 𝑡>𝑀𝑎𝑥_𝑔𝑒𝑛.